A function 'continous' at a 'point'.

A function f(x) is continuous at x-c if corresponding to any positive number ε, arbitrarily assigned, there exists a positive number δ such that -

|f(c+h) - f(c)| < ε

for all values such that |h|<δ

This means that f(c+h) lies between f(c) - ε and f(c) + ε for all values of h lying between -δ and δ.

I was wondering that the continuity of a function is an actually function of these δs...if they are large they might cover values where the function is broken...so things actually depend on these δs and not c if δ is not infinitely small.

if you want to examine a limit as x->a, f(a) is the limit, so you want to pick an interval of points around that limit.. say, an interval of 1. and then there will exist corresponding x values for that interval. if you picked the delta first, who knows what you'll get?

if you ever think about how "for every epsilon there exists a delta" works, think about a function every y there exists x such that ƒ(x) = y

is that when you are "really close" to c, function values are "really close" to f(c).

In general (not always) the smaller the value of [tex] \epsilon [/tex] you select, the smaller must be [tex] \delta [/tex]

Think about this geometric approach. Draw a portion of an arbitrary continuous function (draw any continuous curve) - for a specific example, draw it near [tex] c =2 [/tex], and suppose [tex] f(2) = 5 [/tex].

Now pick [tex] \epsilon = 0.05 [/tex] and draw the two horizontal lines [tex] y = 5 - \epsilon[/tex] and [tex] y = 5 + \epsilon [/tex].
Now draw two vertical lines, one on each side of [tex] c = 2[/tex] (equally spaced) so that all of the graph between these two lines is between the two horizontal lines you drew at the first step. The common distance these vertical lines are from 2 is [tex] \delta [/tex] - so, given [tex] \epsilon = .05 [/tex], you've just found a [tex] \delta > 0 [/tex] such that

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if you want to examine a limit as x->a, f(a) is the limit, so you want to pick an interval of points around that limit.. say, an interval of 1. and then there will exist corresponding x values for that interval. if you picked the delta first, who knows what you'll get?

if you ever think about how "for every epsilon there exists a delta" works, think about a function every y there exists x such that ƒ(x) = y

I absolutely do not know limits...but I did understand what you said (sorta).

I find the variable δ very reluctant...why not skimpily manipulate the value of h?...why make the relation h = δ and THEN manipulate h?

So over all this |f(c+h) - f(c)| < ε && |h|<δ mechanism is trying to say that for an infinitely small change in value of h in the function f(x+h); the value of f(x+h) will also change by an infinity small value within the interval [a, b] (corresponding to value x+h) if the function is called continuous within [a, b]...it's a test for continuity.

By this we can also define the continuity at the 'point' f(x+h) cause here the value of h is infinitely small...so WHY h, δ, x, ε........SO many variables?

Anyway...I do not even understand the definition of the variables ε...it's neither f(x) nor f(x-h)...what is it (theoretically)?

^
wow!
Using δ instead of h has three purposes
1)Greek letters are really cool and fun to write.
2)δ>|h| so δ is a ristriction on h not its value
3)δ establishes a short hand that, one knows what it means in different situations.

Infinitely small is a strange term, we have a better one infinitesimal.
h, δ, ε are note infinitesimal they are small enough.
In words the definition of a limit L=lim_x->a f(x) would be
"The function f can be made as close to L as desired by chosing x sufficiently close to a"
or for f is continuous at a
"The function f can be made as close to f(a) as desired by chosing x sufficiently close to a"

ε the maximum allowed difference between the function and the limit

Think of it like a traffic law. Your driving around in you facy function f and you see a sign "|f(x+h)-f(x)|<ε". If f is contimuous you can manage this by making sure |h|<δ so you will not get a ticket.

If I give you some small value epsilon, you can give me a small value of delta so that if y is within delta of x, f(y) is within epsilon of f(x). It's like a game... your goal is to show something is continuous by always giving me a delta back when I give you epsilon

h – A value added/subtracted to x; since it will be added/subtracted to x, f(x + h) or f(x – h), it will return a deviated value as compared to f(x). It's a criteria that the deviation should be < ε...in mathematical terms |f(x+h)-f(x)| < ε or |f(x-h)-f(x)| < ε. So actually this value (h) has to be computed keeping the value of ε in mind.

A function is said to be continuous iff for a very small value of ε the corresponding value of h is also small; notice that by this definition, even if |f(x+h)-f(x)| = 0, the function will be considered continuous.
If taking a fresh example of the function g(e) = y; for an infinitely small change in value of e, there should be an infinitely small change or no change in value of y for the function to be continuous.

Most probably this is wrong cause I found the need of another variable, δ unnecessary.

h is not computed with \epsilon in mind. h is arbitrary, what we are specifying is that there is a continuous range from -h to h inclusive by which the distance of f(x+h) from f(x) is always less than \epsilon. We are interested in the behavior of the function locally about x, we do not care what h is, we only care about the restrictions on h and the restrictions on the change in the function when we deviate by some h.

h is not computed with \epsilon in mind. h is arbitrary, what we are specifying is that there is a continuous range from -h to h inclusive by which the distance of f(x+h) from f(x) is always less than \epsilon. We are interested in the behavior of the function locally about x, we do not care what h is, we only care about the restrictions on h and the restrictions on the change in the function when we deviate by some h.

Thanks A LOT man...that helped my by A LOT...finally something clear that I can understand.

So we can say that |f(x+h) - f(x)| and |f(x-h) - f(x)| should not exceed ε and ε is the constant here. Since |f(x+h) - f(x)| and |f(x-h) - f(x)| is also a function of h, ε poses a restriction on h.

But this is sorta defining the limit...not continuity; what do we say for continuity?